Question

1. Use multiple regression analysis to study the variation in mercury concentration in largemouth bass living in Florida lakes. The data (bass.csv on Canvas) come from a study of 53 lakes in Florida sampled from the summer of 1990 to the spring of 1991 (Lange, Royals, and Connor 1993). During this time, samples of water were taken from the lakes and the follows factors were measured: pH, alkalinity, amount of chlorophyll from suspended plant matter, and the concentration of calcium. At the same time, fish were caught, and their flesh was tested for mercury levels. The response variable is the average mercury level in the flesh of bass in each of the 53 lakes (avg_mercury). You will use this data to determine if the level of mercury in the fish can be predicted based on the water chemistry. Import the data and fit the multiple regression model, fitting all possible models.

#question1
library("olsrr")

##
## Attaching package: 'olsrr'

## The following object is masked from 'package:datasets':
##
##     rivers

bass = read.csv("bass.csv", header = T)
attach(bass)
bass.lm = lm(Avg_Mercury ~ Alkalinity + pH + Calcium + Chlorophyll, data = bass)
bass.all= ols_step_all_possible(bass.lm)
bass.all

##    Index N                        Predictors  R-Square Adj. R-Square
## 1      1 1                        Alkalinity 0.4254905     0.4142256
## 2      2 1                                pH 0.3310853     0.3179693
## 3      3 1                           Calcium 0.2386129     0.2236838
## 4      4 1                       Chlorophyll 0.2130176     0.1975865
## 6      5 2                Alkalinity Calcium 0.4478582     0.4257726
## 7      6 2            Alkalinity Chlorophyll 0.4436411     0.4213868
## 5      7 2                     Alkalinity pH 0.4292584     0.4064287
## 9      8 2                    pH Chlorophyll 0.3444788     0.3182580
## 8      9 2                        pH Calcium 0.3348995     0.3082955
## 10    10 2               Calcium Chlorophyll 0.3009248     0.2729618
## 13    11 3    Alkalinity Calcium Chlorophyll 0.4705171     0.4380997
## 11    12 3             Alkalinity pH Calcium 0.4576077     0.4244001
## 12    13 3         Alkalinity pH Chlorophyll 0.4436478     0.4095855
## 14    14 3            pH Calcium Chlorophyll 0.3484270     0.3085347
## 15    15 4 Alkalinity pH Calcium Chlorophyll 0.4719492     0.4279450
##    Mallow's Cp
## 1     3.223111
## 2    11.804576
## 3    20.210347
## 4    22.536973
## 6     3.189877
## 7     3.573211
## 5     4.880607
## 9    12.587099
## 8    13.457860
## 10   16.546176
## 13    3.130182
## 11    4.303642
## 12    5.572602
## 14   14.228213
## 15    5.000000

plot(bass.all)

detach(bass)

a. Give the ??2 and Adjusted??2 for the best models with one, two, three, and four predictors. Comment on these results (include the variables involved.)

b. Suppose that you want to predict the average mercury level of fish in a new lake with alkalinity 3.0, calcium 2.5, chlorophyll 2.5, and pH 6.0. The predicted value for the model including all four predictors is .545 (.0164, 1.073) [mean (PI).] The predicted value for the model including only alkalinity, calcium, and chlorophyll is .532 (.0133, 1.051). Have the predicted values and the prediction intervals changed considerably between the two models? Explain why or why not (based on the inspection of these results.)

c. Explain how your results of a) and b) agree.

Answer 1

a)

The ?2 and Adjusted ?2 for the best models with one predictors is

Alkalinity 0.4254905     0.4142256

The ?2 and Adjusted ?2 for the best models with two predictors is

Alkalinity Calcium 0.4478582     0.4257726

The ?2 and Adjusted ?2 for the best models with three predictors is

Alkalinity Calcium Chlorophyll  0.4705171     0.4380997

The ?2 and Adjusted ?2 for the best models with four predictors is

Alkalinity pH Calcium Chlorophyll 0.4719492     0.4279450

The Adjusted ?2 of models with three predictors (Alkalinity, Calcium, Chlorophyll ) is the highest. So, the best models to predict mercury levels is model with predictors Alkalinity, Calcium, Chlorophyll.

b)

The predicted value for the model including all four predictors is .545 (.0164, 1.073)

The predicted value for the model including only alkalinity, calcium, and chlorophyll is .532 (.0133, 1.051)

The prediction intervals for both the models overlap. Thus, the predicted values and the prediction intervals does not changed considerably between the two models.

c.

As per results in part (a), the ?2 of model with three predictors (Alkalinity, Calcium, Chlorophyll ) and model with four predictors (Alkalinity, pH, Calcium, Chlorophyll ) are almost same. This is in agreement to part (b), where the prediction intervals of model with three and four predictors are not significantly different.